
# q-expansion of newform 1152.2.a.f, downloaded from the LMFDB on 19 April 2024.

# We generate the q-expansion using the Hecke eigenvalues a_p at the primes.
# Each a_p is given as a linear combination
# of the following basis for the coefficient ring.

def make_data():

    from sage.all import prod, floor, prime_powers, gcd, QQ, primes_first_n, next_prime, RR

    def discrete_log(elts, gens, mod):
        # algorithm 2.2, page 16 of https://arxiv.org/abs/0903.2785
        def table_gens(gens, mod):
            T = [1]
            n = len(gens)
            r = [None]*n
            s = [None]*n
            for i in range(n):
                beta = gens[i]
                r[i] = 1
                N = len(T)
                while beta not in T:
                    for Tj in T[:N]:
                        T.append((beta*Tj) % mod)
                    beta = (beta*gens[i]) % mod
                    r[i] += 1
                s[i] = T.index(beta)
            return T, r, s
        T, r, s = table_gens(gens, mod)
        n = len(gens)
        N = [ prod(r[:j]) for j in range(n) ]
        Z = lambda s: [ (floor(s/N[j]) % r[j]) for j in range(n)]
        return [Z(T.index(elt % mod)) for elt in elts]
    def extend_multiplicatively(an):
        for pp in prime_powers(len(an)-1):
            for k in range(1, (len(an) - 1)//pp + 1):
                if gcd(k, pp) == 1:
                    an[pp*k] = an[pp]*an[k]
    from sage.all import PolynomialRing, NumberField
    R = PolynomialRing(QQ, "x")
    f = R(poly_data)
    K = NumberField(f, "a")
    betas = [K.gens()[0]**i for i in range(len(poly_data))]
    convert_elt_to_field = lambda elt: sum(c*beta for c, beta in zip(elt, betas))
    # convert aps to K elements
    primes = primes_first_n(len(aps_data))
    good_primes = [p for p in primes if not p.divides(level)]
    aps = map(convert_elt_to_field, aps_data)
    if not hecke_ring_character_values:
        # trivial character
        char_values = dict(zip(good_primes, [1]*len(good_primes)))
    else:
        gens = [elt[0] for elt in hecke_ring_character_values]
        gens_values = [convert_elt_to_field(elt[1]) for elt in hecke_ring_character_values]
        char_values = dict([(
            p,prod(g**k for g, k in zip(gens_values, elt)))
            for p, elt in zip(good_primes, discrete_log(good_primes, gens, level))
            ])
    an_list_bound = next_prime(primes[-1])
    an = [0]*an_list_bound
    an[1] = 1
    
    from sage.all import PowerSeriesRing
    PS = PowerSeriesRing(K, "q")
    for p, ap in zip(primes, aps):
        if p.divides(level):
            euler_factor = [1, -ap]
        else:
            euler_factor = [1, -ap, p**(weight - 1) * char_values[p]]
        k = RR(an_list_bound).log(p).floor() + 1
        foo = (1/PS(euler_factor)).padded_list(k)
        for i in range(1, k):
            an[p**i] = foo[i]
    extend_multiplicatively(an)
    return PS(an)
level = 1152
weight = 2
poly_data = [0, 1]

# The basis for the coefficient ring is just the power basis
# in the root of the defining polynomial above.
hecke_ring_character_values = None
aps_data = [[0], [0], [-2], [2], [-4], [-2], [4], [4], [-8], [-6], [-6], [2], [12], [-12], [-8], [-6], [-8], [10], [8], [0], [2], [-14], [-12], [-8], [-2], [-6], [14], [8], [-18], [8], [-2], [0], [4], [-8], [6], [10], [18], [4], [-8], [6], [24], [-2], [8], [-2], [26], [-10], [-8], [-18], [12], [14], [-24], [24], [-14], [12], [0], [0], [26], [14], [-18], [0], [0], [-22], [-24], [-16], [22], [18], [16], [10], [-12], [26], [24], [32], [10], [-22], [20], [-24], [14], [-30], [12], [-26], [-20], [14], [-8], [-30], [6], [-12], [-28], [6], [-34], [-2], [-4], [0], [-38], [-32], [-32], [-16], [2], [-4], [4], [38], [-4], [-2], [-36], [-4], [40], [-10], [-32], [24], [-24], [-18], [14], [-22], [16], [-16], [-38], [36], [12], [-24], [-18], [48], [-22], [-14], [-26], [-36], [36], [10], [-2], [48], [18], [6], [-24], [-40], [26], [-2], [4], [14], [-6], [-20], [-42], [-12], [12], [2], [26], [8], [14], [0], [10], [-36], [-4], [-8], [10], [32], [20], [48], [-20], [24], [-6], [-36], [-38], [14], [8], [-20], [22], [28], [12], [-16], [58], [18], [-34], [14], [48], [-30], [40], [38], [10], [-48], [-48], [-26], [38], [10], [-14], [-12], [6], [-32], [-24], [30], [-26], [64], [10], [16], [46], [-24], [-16], [22], [-16], [32], [-58], [-10], [32], [0], [-2], [-10], [-58], [-38], [-12], [-54], [-50], [-4], [-40], [44], [62], [-58], [70], [8], [48], [-38], [-6], [4], [-24], [30], [22], [-18], [-36], [6], [-28], [10], [56], [-16], [22], [32], [46], [-20], [14], [-32], [-68], [-32], [14], [-26], [12], [-72], [-36], [16], [34], [2], [64], [64], [26], [-24], [32], [0], [-66], [40], [-56], [22], [-46], [-56], [70], [72], [-30], [-6], [-10], [-8], [-26], [-62], [12], [48], [-10], [72], [-4], [-74], [58], [-56], [-26], [50], [-18], [62], [4], [-30], [30], [-72], [-16], [-58], [24], [-22], [-52], [-32], [-14], [30], [-14], [40], [-34], [24], [-24], [-72], [26], [30], [22], [-70], [36], [-4], [-10], [-42], [-6], [48], [68], [-30], [-4], [86], [48], [-70], [40], [-54], [-12], [-28], [-16], [-42], [-32], [64], [-6], [-40], [-32], [-58], [-6], [-70], [-36], [-34], [-60], [-44], [32], [-58], [14], [78], [62], [-24], [56], [-60], [34], [-40], [18], [38], [26], [-52], [-30], [-26], [18], [76], [-74], [48], [-48], [2], [-52], [-10], [-6], [30], [10], [-16], [72], [0], [-60], [48], [50], [-24], [-24], [4], [8], [-82], [-14], [-62], [-22], [-28], [-76], [0], [90], [-62], [-10], [-20], [-24], [-14], [-36], [58], [-58], [-12], [26], [-48], [-4], [24], [10], [-82], [-56], [-56], [46], [30], [-16], [-76], [8], [-74], [22], [76], [-12], [-42], [18], [60], [6], [-36], [-78], [90], [94], [12], [52], [-20], [54], [30], [-4], [-92], [-46], [-14], [40], [-46], [60], [96], [-58], [-6], [32], [12], [102], [-78], [92], [-72], [12], [-24]]
